Solve the Rational Equation for x (x+2)^(4/3)=1/81

${(x + 2)}^{\frac{4}{3}} = \frac{1}{81}$

Raise each side of the equation to the power of $\frac{3}{4}$ to eliminate the fractional exponent on the left side.

${({(x + 2)}^{\frac{4}{3}})}^{\frac{3}{4}} = \pm {(\frac{1}{81})}^{\frac{3}{4}}$

Simplify the exponent.

Simplify the left side.

Simplify ${({(x + 2)}^{\frac{4}{3}})}^{\frac{3}{4}}$ .

Multiply the exponents in ${({(x + 2)}^{\frac{4}{3}})}^{\frac{3}{4}}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(x + 2)}^{\frac{4}{3} \cdot \frac{3}{4}} = \pm {(\frac{1}{81})}^{\frac{3}{4}}$

Cancel the common factor of $4$ .

Cancel the common factor.

${(x + 2)}^{\frac{4}{3} \cdot \frac{3}{4}} = \pm {(\frac{1}{81})}^{\frac{3}{4}}$

Rewrite the expression.

${(x + 2)}^{\frac{1}{3} \cdot 3} = \pm {(\frac{1}{81})}^{\frac{3}{4}}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(x + 2)}^{\frac{1}{3} \cdot 3} = \pm {(\frac{1}{81})}^{\frac{3}{4}}$

Rewrite the expression.

${(x + 2)}^{1} = \pm {(\frac{1}{81})}^{\frac{3}{4}}$

Simplify.

$x + 2 = \pm {(\frac{1}{81})}^{\frac{3}{4}}$

Simplify the right side.

Simplify $\pm {(\frac{1}{81})}^{\frac{3}{4}}$ .

Simplify the expression.

Apply the product rule to $\frac{1}{81}$ .

$x + 2 = \pm \frac{1^{\frac{3}{4}}}{81^{\frac{3}{4}}}$

One to any power is one.

$x + 2 = \pm \frac{1}{81^{\frac{3}{4}}}$

Simplify the denominator.

Rewrite $81$ as $3^{4}$ .

$x + 2 = \pm \frac{1}{{(3^{4})}^{\frac{3}{4}}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x + 2 = \pm \frac{1}{3^{4 (\frac{3}{4})}}$

Cancel the common factor of $4$ .

Cancel the common factor.

$x + 2 = \pm \frac{1}{3^{4 (\frac{3}{4})}}$

Rewrite the expression.

$x + 2 = \pm \frac{1}{3^{3}}$

Raise $3$ to the power of $3$ .

$x + 2 = \pm \frac{1}{27}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x + 2 = \frac{1}{27}$

Move all terms not containing $x$ to the right side of the equation.

Subtract $2$ from both sides of the equation.

$x = \frac{1}{27} - 2$

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{27}{27}$ .

$x = \frac{1}{27} - 2 \cdot \frac{27}{27}$

Combine $- 2$ and $\frac{27}{27}$ .

$x = \frac{1}{27} + \frac{- 2 \cdot 27}{27}$

Combine the numerators over the common denominator.

$x = \frac{1 - 2 \cdot 27}{27}$

Simplify the numerator.

Multiply $- 2$ by $27$ .

$x = \frac{1 - 54}{27}$

Subtract $54$ from $1$ .

$x = \frac{- 53}{27}$

Move the negative in front of the fraction.

$x = - \frac{53}{27}$

Next, use the negative value of the $\pm$ to find the second solution.

$x + 2 = - \frac{1}{27}$

Move all terms not containing $x$ to the right side of the equation.

Subtract $2$ from both sides of the equation.

$x = - \frac{1}{27} - 2$

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{27}{27}$ .

$x = - \frac{1}{27} - 2 \cdot \frac{27}{27}$

Combine $- 2$ and $\frac{27}{27}$ .

$x = - \frac{1}{27} + \frac{- 2 \cdot 27}{27}$

Combine the numerators over the common denominator.

$x = \frac{- 1 - 2 \cdot 27}{27}$

Simplify the numerator.

Multiply $- 2$ by $27$ .

$x = \frac{- 1 - 54}{27}$

Subtract $54$ from $- 1$ .

$x = \frac{- 55}{27}$

Move the negative in front of the fraction.

$x = - \frac{55}{27}$

The complete solution is the result of both the positive and negative portions of the solution.

$x = - \frac{53}{27}, - \frac{55}{27}$

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{53}{27}, - \frac{55}{27}$

Decimal Form:

$x = - 1.\overline{962}, - 2.\overline{037}$

Mixed Number Form:

$x = - 1 \frac{26}{27}, - 2 \frac{1}{27}$

Solve the Rational Equation for x (x+2)^(4/3)=1/81

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